The touching and splitting of the phonon bands near the
critical magnetic field induces the abrupt change of Chern
numbers of the phonon band [1]}. In Ref. [1]}, for the PHE
in the honeycomb lattices, we know
that band 2 and 3 are going to touch with each other at the \({\bf \Gamma }\) 
point if the magnetic field increases to \(h_c\) ; at the critical
magnetic field, the degeneracy occurs and the two bands possess the
cone shape; above the critical point \(h_c\) , the two bands split up.
Therefore, the difference between the two bands decreases below and
increases above the critical magnetic field,
and is zero at the critical point. The eigenfrequecy difference is in the
denominator of the Berry curvature, thus the variation of the difference around
the critical magnetic field, dramatically affects the Berry curvature of the
corresponding bands. In the kagome lattice systems,
we find that the touching and splitting of the phonon bands not only occurs at
the \({\bf \Gamma }\)  point, but also occurs at other points, which is shown in
Fig. REF . At the first
critical points \(h_{c1}\) , the bands 2 and 3 touch at the point \({\bf K}\) 
(marked by a square with number 1); at \(h_{c2}\)  the two bands touch at \({\bf X}\) 
( marked by a square with number 2); while only for the third
critical one \(h_{c3}\) , band 2 and 3 degenerate at the point \({\bf \Gamma }\) 
(marked by a square with number 3).
From the contour maps of bands 2 and 3, we clearly see that the critical
magnetic fields \(h_{c1}\) , \(h_{c2}\) , and \(h_{c3}\) , there are local maximum
for the band 2 and the local minimum for the band 3. Thus for all
the critical magnetic fields where the Chern numbers have abrupt changes, in the wave-vector space we can always find the phonon bands touching
and splitting at some symmetric center points.
